>>
<<
Ndx
Usr
Pri
JfC
LJ
Phr
Dic
Rel
Voc
!:
wd
Help
Dictionary
Determinant 
u . v 2 _ _ 
Dot Product

The phrases / . * and +/ . * are the determinant
and permanent of square matrix arguments. More generally,
the phrase u . v is defined in terms of a recursive expansion
by minors along the first column, as discussed below.


For vectors and matrices, the phrase x +/ . * y
is equivalent to the dot, inner, or matrix
product of math; other rank0 verbs such as <. and *.
are treated analogously. In general, u . v is defined
by u@(v"(1+lv,_)) , restated in English below.

For example:
x=: 1 2 3 [ m=: >1 6 4;4 1 0;6 6 8
det=: / . *
mp=: +/ . *
x ([ ; ] ; det@] ; mp ; mp~ ; mp~@]) m
+++++++
1 2 31 6 4_11227 26 2825 6 4249 36 36
 4 1 0    8 25 16
 6 6 8   78 90 88
+++++++
The monad u . v is defined as illustrated below:
DET=: 2 : 'v/@,`({."1 u . v $:@minors)@.(0<{:@$) @ ,. "2'
minors=: }."1 @ (1&([\.))
/ DET * m
_112
/ DET * 1 16 64
49
/ DET * i.3 0
1
+/ DET * m
320
The definition u@(v"(1+lv,_)) given above for the dyadic
case may be restated in words as follows: u is applied to
the result of v on lists of “left argument cells”
and the right argument in toto. The number of items in a
list of left argument cells must agree with the number in the right argument.
Thus, if v has ranks 2 3 and the shapes
of x and y are 2 3 4 5 6
and 4 7 8 9 10 11, then there are 2 3 lists
of left argument cells (each shaped 4 5 6); and if
the shape of a result cell is sr, the overall shape
is 2 3,sr .
>>
<<
Ndx
Usr
Pri
JfC
LJ
Phr
Dic
Rel
Voc
!:
wd
Help
Dictionary